Question

Решить : 5^x-3^x<3^(x+1)-7*5^(x-1)

Answer 1

5ˣ-3ˣ < 3ˣ ⁺¹ - 7*5ˣ⁻¹

5ˣ -3ˣ < 3*3ˣ - 7*5ˣ/5

5ˣ + 7*5ˣ/5 < 3*3ˣ +3ˣ

(5* 5ˣ + 7*5ˣ)/5 < 4*3ˣ

12*5ˣ < 20*3ˣ

20*3ˣ /12*5ˣ > 1

(5/3)*(3/5)ˣ > 1

(3/5)ˣ > 3/5

так как 3/5 <1, x< 1

x∈(-∞; 1)

Answer 2

We can start by simplifying the right-hand side of the inequality:

3^(x+1) - 75^(x-1) = 33^x - 7*(5^2 * 5^(-1))^x = 33^x - 7(25/5)^x = 33^x - 75^x

Substituting this expression back into the original inequality, we get:

5^x - 3^x < 33^x - 75^x

Now we can simplify both sides by collecting like terms:

5^x - 75^x < 3^x - 33^x -65^x < -23^x Dividing both sides by 2*5^x (which is positive), we get:

-3/5 < (3/5) * (3/2)^x

Taking the logarithm base 3/2 of both sides, we get:

log(3/2)(-3/5) < x

Using a calculator, we can find that log(3/2)(-3/5) ≈ -1.843. Therefore, the solution to the inequality is:

x > -1.843

So the set of real numbers that satisfy the inequality is:

{x | x > -1.843}